In this chapter we introduce the theory of Diophantine approximation via a series of basic examples from information theory relevant to wireless communications. In particular, we discuss Dirichlet's theorem, badly approximable points, Dirichlet improvable and singular points, the metric (probabilistic) theory of Diophantine approximation including the Khintchine-Groshev theorem and the theory of Diophantine approximation on manifolds. We explore various number theoretic approaches used in the analysis of communication characteristics such as Degrees of Freedom (DoF). In particular, we improve the result of Motahari et al regarding the DoF of a two-user X-channel. In essence, we show that the total DoF can be achieved for all (rather than almost all) choices of channel coefficients with the exception of a subset of strictly smaller dimension than the ambient space. The improvement utilises the concept of jointly non-singular points that we introduce and a general result of Kadyrov et al on the $\delta$-escape of mass in the space of lattices. We also discuss follow-up open problems that incorporate a breakthrough of Cheung and more generally Das et al on the dimension of the set of singular points.

中文翻译：

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数论遇到无线通信：像我们这样的傻瓜的介绍

在本章中，我们通过与无线通信相关的信息论中的一系列基本示例来介绍丢番图近似理论。特别地，我们讨论了狄利克雷定理、极近似点、狄利克雷可改进点和奇异点、丢番图逼近的度量（概率）理论，包括 Khintchine-Groshev 定理和流形上的丢番图逼近理论。我们探索了用于分析通信特征（例如自由度 (DoF)）的各种数论方法。特别是，我们改进了 Motahari 等人关于双用户 X 通道的 DoF 的结果。在本质上，我们表明，对于所有（而不是几乎所有）信道系数的选择，除了一个维度比环境空间更小的子集外，可以实现总的自由度。改进利用了我们引入的联合非奇异点的概念和 Kadyrov 等人关于格空间中质量的 $\delta$-escape 的一般结果。我们还讨论了后续开放问题，这些问题结合了 Cheung 的突破，更一般地说是 Das 等人在奇异点集的维度上的突破。